12/06/2013 – 10/07/2013An Introduction to Quantum Logic and Quantum Probabilities by Federico Holik
The logic-algebraic approach to quantum mechanics provides a formal framework in which almost all foundational and interpretational problems can be posed and treated in a rigorous way. Its scope is not restricted to non-relativistic quantum mechanics, but can be extended to relativistic quantum field theory providing an axiomatic formulation. It also provides a natural formal framework for the study of a generalized probability theory, which includes Kolmogorovian and quantum probabilities as particular cases of a vast family. In this seminar we will provide an introduction to quantum logic and the algebraic approach to quantum mechanics. Projective valued measures (PVMs) and positive operator valued measures (POVMs) will be discussed, followed by a review of elementary notions of lattice theory. Next, the relationship between lattice theory and von Neumann algebras will be discussed in connection to the problems posed by von Neumann regarding quantum probabilities. In particular, we will discuss non commutative probability theory and the convex operational approach to statistical theories. The seminar will consist of 5 sessions. The above mentioned contents will be covered in first three sessions, while the 4th and 5th are left open to discuss topics inspired by the questions and problems which emerge from the three former ones.

13/03/2013 – 29/03/2013 An Introduction to Canonical Quantum Gravityby Federico Zalamea
Canonical Quantum Gravity relies mainly on two ideas: on the one hand, the fact that fundamental object of General Relativity can be considered to be a connexion rather than a metric, establishing thus a close analogy with Yang-Mills theories ; on the other hand, Dirac’s canonical quantization program for constrained Hamiltonian systems (called Refined Algebraic Quantization in its modern version). The goal of this lectures will be to present the first steps in the construction of the theory, paying special attention to the underlying mathematical structures and to the concepts that seem to be most important. More in detail, we will first show how Ashtekar variables (which constitute the starting point of the theory) can be understood in the light of Cartan’s theory of connexions. Then, we will apply Dirac’s procedure in order to build the quantum states space and we will introduce the notion of a spin network.

13/02/2013 – 06/03/2013Clifford Algebras and Spinors by Corentin Le Fur
Spinors are complex mathematical entities and play an important role in mathematical and theoretical physics. In their mathematical form, they were introduced by Cartan in 1913. The word « Spinor » seems to have been used for the first time by Ehrenfest and Wolfgang Pauli inaugurated its use in mathematical physics in 1927.
This course has two main goals : on the one hand, to build in a mathematical and rigorous way the notion of spinor and, on the other hand, inspire a conceptual and philosophical analysis of this notion.
We will begin by the study of Clifford Algebras defined by quadratic forms and will examine a few particular cases. Then, we will look at orthogonal and spin groups which will lead us to the concept of spinors space. Fundamental algebraic structures (as monoid, group, ring, algebra, field...) are considered as known.

18/01/2013 – 06/02/2013An Introduction to Hopf Algebras by Julien Page
Hopf Algebras define a very rich algebraic structure, playing a crucial role in many fields of mathematics and mathematical physics, specially in non-commutative geometry. We will show how this notion solves a duality default of the category of finite groups by extending the Pontryagin duality of the category of abelian finite groups. Next, we will show how this algebraic structure generalizes the structure of a group and its concomitant notion : symmetry. (As an algebraic structure, it is a generalization by the existence of a generalized notion of an inverse, by its "actions" on mathematical objects, by the "integrals" it helps to define, etc). Finally, we will make use of this structure to analyze the notion of duality, focusing mainly in the (crucial for physics) duality between space of states and algebra of observables.
We will suppose that the following mathematical notions are already familiar to the audience: group, ring, field, module, vector space, algebra, tensorial product, category, functor.

Bibliography: to start, we will follow the notes downloadable here (in French) : http://math.univ-bpclermont.fr/ bic... and we will refer to the beginning of Shahn Majid’s book : Foundations of Quantum Group Theory, Cambridge University Press, 2000

12/12/2012–09/01/2013 The Theory of Cartan Connections by Gabriel Catren
While Ehresmann connections are the geometric object used to represent the "gauge fields" of the Yang-Mills theory (giving a geometric description of the electromagnetic and nuclear interactions), the Cartan connections enable a reformulation (and a generalization) of General Relativity in such a way that the fundamental variable of the theory will no longer be the space-time metric but will be a connection instead. This reformulation gives new perspectives to the program trying to understand General Relativity as a "gauge theory", i.e. as a theory describing a dynamical connection on a fiber bundle over space-time. We will show in which precise manner General Relativity (and its generalization : the Einstein-Cartan theory) is the result of the "locality" of the affine group acting transitively on the vacuum solution of the theory (Poincaré, de Sitter or anti-de Sitter groups)

This project has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) for research, technological development and demonstration under grant agreement n° 263523 (Project Philosophy of Canonical Quantum Gravity)